research-article

On the convergence of the Hegselmann-Krause system

Authors:

Arnab Bhattacharyya,

Mark Braverman,

Bernard Chazelle,

Huy L. NguyenAuthors Info & Claims

ITCS '13: Proceedings of the 4th conference on Innovations in Theoretical Computer Science

Pages 61 - 66

https://doi.org/10.1145/2422436.2422446

Published: 09 January 2013 Publication History

Abstract

We study convergence of the following discrete-time non-linear dynamical system: $n$ agents are located in R^d and at every time step, each moves synchronously to the average location of all agents within a unit distance of it. This popularly studied system was introduced by Krause to model the dynamics of opinion formation and is often referred to as the Hegselmann-Krause model. We prove the first polynomial time bound for the convergence of this system in arbitrary dimensions. This improves on the bound of n^O(n) resulting from a more general theorem of Chazelle [4]. Also, we show a quadratic lower bound and improve the upper bound for one-dimensional systems to O(n³).

References

[1]

R. Axelrod. The dissemination of culture: a model with local convergence and global polarization. Journal of Conflict Resolution, 1997. Reprinted in The complexity of cooperation," Princeton University Press, Princeton, 1997.

[2]

V. D. Blondel, J. M. Hendricx, and J. N. Tsitsiklis. On the 2R conjecture for multi-agent systems. In European Control Conference, pages 2996--3000, July 2007.

[3]

V. D. Blondel, J. M. Hendricx, and J. N. Tsitsiklis. On Krause's multi-agent consensus model with state-dependent connectivity. IEEE Transactions on Automatic Control, 54(11), Nov. 2009.

[4]

B. Chazelle. The total s-energy of a multiagent system. SIAM J. Control and Optimization, 49(4):1680--1706, 2011.

[5]

B. Chazelle. The dynamics of influence systems. arXiv:1204.3946v2, 2012. Prelim. version in Proc. 53rd FOCS, 2012.

Digital Library

[6]

B. Chazelle. Natural algorithms and influence systems. CACM Research Highlights, 2012.

Digital Library

[7]

D. A. Easley and J. M. Kleinberg. Networks, Crowds, and Markets - Reasoning About a Highly Connected World. Cambridge University Press, 2010.

Digital Library

[8]

S. Fortunato. On the consensus threshold for the opinion dynamics of krause-hegselmann. International Journal of Modern Physics C, 16(2):259--270, 2005.

[9]

R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: models, analysis and simulation. J. Artificial Societies and Social Simulation, 5(3), 2002.

[10]

J. M. Hendrickx and V. D. Blondel. Convergence of different linear and non-linear Vicsek models. In Proc. 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS2006), pages 1229--1240, July 2006.

[11]

U. Krause. A discrete nonlinear and non-autonomous model of consensus formation. In Proc. Commun. Difference Equations, pages 227--236, 2000.

[12]

J. Lorenz. A stabilization theorem for dynamics of continuous opinions. Physica A: Statistical Mechanics and its Applications, 355(1):217--223, 2005.

[13]

J. Lorenz. Continuous opinion dynamics under bounded confidence: a survey. International Journal of Modern Physics C, 16(18):1819--1838, 2007.

[14]

S. Martinez, F. Bullo, J. Cortes, and E. Frazzoli. On synchronous robotic networks--Part ii: Time complexity of rendezvous and deployment algorithms. IEEE Transactions on Automatic Control, 52(12):2214--2226, Dec. 2007.

[15]

A. Mirtabatabaei and F. Bullo. Opinion dynamics in heterogeneous networks: convergence, conjectures and theorems. arXiv:1103.2829v2, Mar. 2011. To appear in SIAM J. Control Optim.

[16]

L. Moreau. Stability of multiagent systems with time-dependent communication links. IEEE Transactions on Automatic Control, 50(2):169--182, Feb. 2005.

[17]

M. Roozbehani, A. Megretski, and E. Frazzoli. Lyapunov analysis of quadratically symmetric neighborhood consensus algorithms. In CDC, pages 2252--2257, 2008.

[18]

B. Touri and A. Nedic. Discrete-time opinion dynamics. In Signals, Systems and Computers (ASILOMAR), 2011 Conference Record of the Forty Fifth Asilomar Conference on, pages 1172--1176, Nov. 2011.

Cited By

Mei WHendrickx JChen GBullo FDörfler F(2024)Convergence, Consensus, and Dissensus in the Weighted-Median Opinion DynamicsIEEE Transactions on Automatic Control10.1109/TAC.2024.337675269:10(6700-6714)Online publication date: Oct-2024
https://doi.org/10.1109/TAC.2024.3376752
Berenbrink PHoefer MKaaser DLenzner PRau MSchmand D(2024)Asynchronous opinion dynamics in social networksDistributed Computing10.1007/s00446-024-00467-337:3(207-224)Online publication date: 26-Apr-2024
https://doi.org/10.1007/s00446-024-00467-3
Berenbrink PCooper CGava CMarzagão DMallmann-Trenn FRadzik TRivera NOshman RNolin AHalldorsson MBalliu A(2023)Distributed Averaging in Opinion DynamicsProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594593(211-221)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594593
Show More Cited By

Index Terms

On the convergence of the Hegselmann-Krause system
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations

There are several forms of systems of nonsmooth equations which are equivalent to the Karush-Kuhn-Tucker KKT system of a nonlinearly constrained optimization problem NLP. If the NLP is twice continuously differentiable and the Hessian functions of its ...
Opinion-Climate-Based Hegselmann-Krause dynamics
Highlights
- A novel opinion-climate-based Hegselmann–Krause dynamics model for Cyber-Physical-Social Services is proposed.
- The opinion climate is quantified by calculating the relative proportion of agents who hold positive or negative opinion.
Abstract
Opinion dynamics, which focuses on the opinion evolution process of a group of agents, is an important problem in Cyber-Physical-Social Systems (CPSS). Although the classic Hegselmann-Krause model and its variants have been intensively studied, ...
On the convergence of descent methods for monotone variational inequalities

Recently, Zhu and Marcotte (1993) established the convergence of a modified descent algorithm for monotone variational inequalities. Using algorithmic equivalence results due to Patriksson (1993) and Larsson and Patriksson (1994), we show that this ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ITCS '13: Proceedings of the 4th conference on Innovations in Theoretical Computer Science

January 2013

594 pages

ISBN:9781450318594

DOI:10.1145/2422436

Program Chair:
Robert Kleinberg
Cornell University

Copyright © 2013 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 January 2013

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ITCS '13

Sponsor:

SIGACT

ITCS '13: Innovations in Theoretical Computer Science

January 9 - 12, 2013

California, Berkeley, USA

Acceptance Rates

Overall Acceptance Rate 172 of 513 submissions, 34%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

43
Total Citations
View Citations
500
Total Downloads

Downloads (Last 12 months)86
Downloads (Last 6 weeks)9

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Mei WHendrickx JChen GBullo FDörfler F(2024)Convergence, Consensus, and Dissensus in the Weighted-Median Opinion DynamicsIEEE Transactions on Automatic Control10.1109/TAC.2024.337675269:10(6700-6714)Online publication date: Oct-2024
https://doi.org/10.1109/TAC.2024.3376752
Berenbrink PHoefer MKaaser DLenzner PRau MSchmand D(2024)Asynchronous opinion dynamics in social networksDistributed Computing10.1007/s00446-024-00467-337:3(207-224)Online publication date: 26-Apr-2024
https://doi.org/10.1007/s00446-024-00467-3
Berenbrink PCooper CGava CMarzagão DMallmann-Trenn FRadzik TRivera NOshman RNolin AHalldorsson MBalliu A(2023)Distributed Averaging in Opinion DynamicsProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594593(211-221)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594593
Srivastava TBernardo CAltafini CVasca F(2023)Analyzing the effects of confidence thresholds on opinion clustering in homogeneous Hegselmann–Krause models2023 31st Mediterranean Conference on Control and Automation (MED)10.1109/MED59994.2023.10185838(587-592)Online publication date: 26-Jun-2023
https://doi.org/10.1109/MED59994.2023.10185838
Bechihi APanteley EDuhamel PBouttier A(2023)A Resource Allocation Algorithm for Formation Control of Connected VehiclesIEEE Control Systems Letters10.1109/LCSYS.2022.31878247(307-312)Online publication date: 2023
https://doi.org/10.1109/LCSYS.2022.3187824
Garg SShiragur KGordon DCharikar M(2023)Distributed algorithms from arboreal ants for the shortest path problemProceedings of the National Academy of Sciences10.1073/pnas.2207959120120:6Online publication date: 30-Jan-2023
https://doi.org/10.1073/pnas.2207959120
Continelli EPignotti C(2023)Consensus for Hegselmann–Krause type models with time variable time delaysMathematical Methods in the Applied Sciences10.1002/mma.959946:18(18916-18934)Online publication date: 2-Aug-2023
https://doi.org/10.1002/mma.9599
Li H(2022)Mixed Hegselmann-Krause Dynamics ⅡDiscrete and Continuous Dynamical Systems - B10.3934/dcdsb.2022200(0-0)Online publication date: 2022
https://doi.org/10.3934/dcdsb.2022200
Wang C(2022)Opinion Dynamics with Higher-Order Bounded ConfidenceEntropy10.3390/e2409130024:9(1300)Online publication date: 14-Sep-2022
https://doi.org/10.3390/e24091300
Parasnis RFranceschetti MTouri B(2022)On the Convergence Properties of Social Hegselmann–Krause DynamicsIEEE Transactions on Automatic Control10.1109/TAC.2021.305274867:2(589-604)Online publication date: Feb-2022
https://doi.org/10.1109/TAC.2021.3052748
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents